1. We are asked to simplify the following quadratic expressions: (4) \quad 3x^2 + 7xy + 2y^2 + 11x + 7y + 6 (5) \quad 2x^2 - 7xy + 3y^2 - 9x + 7y + 4 (6) \quad 2x^2 - 8xy + 6y^2 - 11x + 13y + 5 (7) \quad 3x^2 - 7xy + 2y^2 - 11x + 7y + 6 2. These are quadratic expressions in two variables $x$ and $y$. We will check if they can be factored or simplified further. 3. For each expression, we look for factorization or simplification: (4) $3x^2 + 7xy + 2y^2 + 11x + 7y + 6$ Try to factor as $(ax + by + c)(dx + ey + f)$: Assuming $(3x + 2y + 3)(x + y + 2)$: Expanding: $$3x \cdot x = 3x^2$$ $$3x \cdot y = 3xy$$ $$3x \cdot 2 = 6x$$ $$2y \cdot x = 2xy$$ $$2y \cdot y = 2y^2$$ $$2y \cdot 2 = 4y$$ $$3 \cdot x = 3x$$ $$3 \cdot y = 3y$$ $$3 \cdot 2 = 6$$ Sum terms: $$3x^2 + (3xy + 2xy) + 2y^2 + (6x + 3x) + (4y + 3y) + 6 = 3x^2 + 5xy + 2y^2 + 9x + 7y + 6$$ This is close but not exact (we need $7xy$ and $11x$). Try $(3x + y + 2)(x + 2y + 3)$: Expanding: $$3x \cdot x = 3x^2$$ $$3x \cdot 2y = 6xy$$ $$3x \cdot 3 = 9x$$ $$y \cdot x = xy$$ $$y \cdot 2y = 2y^2$$ $$y \cdot 3 = 3y$$ $$2 \cdot x = 2x$$ $$2 \cdot 2y = 4y$$ $$2 \cdot 3 = 6$$ Sum terms: $$3x^2 + (6xy + xy) + 2y^2 + (9x + 2x) + (3y + 4y) + 6 = 3x^2 + 7xy + 2y^2 + 11x + 7y + 6$$ This matches exactly. So (4) factors as: $$(3x + y + 2)(x + 2y + 3)$$ 4. For (5): $2x^2 - 7xy + 3y^2 - 9x + 7y + 4$ Try $(2x - 3y - 1)(x - y - 4)$: Expanding: $$2x \cdot x = 2x^2$$ $$2x \cdot (-y) = -2xy$$ $$2x \cdot (-4) = -8x$$ $$-3y \cdot x = -3xy$$ $$-3y \cdot (-y) = 3y^2$$ $$-3y \cdot (-4) = 12y$$ $$-1 \cdot x = -x$$ $$-1 \cdot (-y) = y$$ $$-1 \cdot (-4) = 4$$ Sum terms: $$2x^2 + (-2xy - 3xy) + 3y^2 + (-8x - x) + (12y + y) + 4 = 2x^2 - 5xy + 3y^2 - 9x + 13y + 4$$ Not exact (we need $-7xy$ and $7y$). Try $(2x - y - 4)(x - 3y + 1)$: Expanding: $$2x \cdot x = 2x^2$$ $$2x \cdot (-3y) = -6xy$$ $$2x \cdot 1 = 2x$$ $$-y \cdot x = -xy$$ $$-y \cdot (-3y) = 3y^2$$ $$-y \cdot 1 = -y$$ $$-4 \cdot x = -4x$$ $$-4 \cdot (-3y) = 12y$$ $$-4 \cdot 1 = -4$$ Sum terms: $$2x^2 + (-6xy - xy) + 3y^2 + (2x - 4x) + (-y + 12y) - 4 = 2x^2 - 7xy + 3y^2 - 2x + 11y - 4$$ Close but not exact. Try $(2x - 7y + 4)(x + y + 1)$: Expanding: $$2x \cdot x = 2x^2$$ $$2x \cdot y = 2xy$$ $$2x \cdot 1 = 2x$$ $$-7y \cdot x = -7xy$$ $$-7y \cdot y = -7y^2$$ $$-7y \cdot 1 = -7y$$ $$4 \cdot x = 4x$$ $$4 \cdot y = 4y$$ $$4 \cdot 1 = 4$$ Sum terms: $$2x^2 + (2xy - 7xy) - 7y^2 + (2x + 4x) + (-7y + 4y) + 4 = 2x^2 - 5xy - 7y^2 + 6x - 3y + 4$$ Not matching. Since factorization is complicated, we leave (5) as is. 5. For (6): $2x^2 - 8xy + 6y^2 - 11x + 13y + 5$ Try factoring out 2 from quadratic terms: $$2(x^2 - 4xy + 3y^2) - 11x + 13y + 5$$ Try $(x - 3y)(x - y) = x^2 - 4xy + 3y^2$ So quadratic part factors as $2(x - 3y)(x - y)$ No simple factorization for linear terms, so expression is: $$2(x - 3y)(x - y) - 11x + 13y + 5$$ 6. For (7): $3x^2 - 7xy + 2y^2 - 11x + 7y + 6$ Try $(3x - 2y - 3)(x - y - 2)$: Expanding: $$3x \cdot x = 3x^2$$ $$3x \cdot (-y) = -3xy$$ $$3x \cdot (-2) = -6x$$ $$-2y \cdot x = -2xy$$ $$-2y \cdot (-y) = 2y^2$$ $$-2y \cdot (-2) = 4y$$ $$-3 \cdot x = -3x$$ $$-3 \cdot (-y) = 3y$$ $$-3 \cdot (-2) = 6$$ Sum terms: $$3x^2 + (-3xy - 2xy) + 2y^2 + (-6x - 3x) + (4y + 3y) + 6 = 3x^2 - 5xy + 2y^2 - 9x + 7y + 6$$ Close but not exact (we need $-7xy$ and $-11x$). Try $(3x - y - 3)(x - 2y - 2)$: Expanding: $$3x \cdot x = 3x^2$$ $$3x \cdot (-2y) = -6xy$$ $$3x \cdot (-2) = -6x$$ $$-y \cdot x = -xy$$ $$-y \cdot (-2y) = 2y^2$$ $$-y \cdot (-2) = 2y$$ $$-3 \cdot x = -3x$$ $$-3 \cdot (-2y) = 6y$$ $$-3 \cdot (-2) = 6$$ Sum terms: $$3x^2 + (-6xy - xy) + 2y^2 + (-6x - 3x) + (2y + 6y) + 6 = 3x^2 - 7xy + 2y^2 - 9x + 8y + 6$$ Close but not exact. Try $(3x - y - 2)(x - 2y - 3)$: Expanding: $$3x \cdot x = 3x^2$$ $$3x \cdot (-2y) = -6xy$$ $$3x \cdot (-3) = -9x$$ $$-y \cdot x = -xy$$ $$-y \cdot (-2y) = 2y^2$$ $$-y \cdot (-3) = 3y$$ $$-2 \cdot x = -2x$$ $$-2 \cdot (-2y) = 4y$$ $$-2 \cdot (-3) = 6$$ Sum terms: $$3x^2 + (-6xy - xy) + 2y^2 + (-9x - 2x) + (3y + 4y) + 6 = 3x^2 - 7xy + 2y^2 - 11x + 7y + 6$$ This matches exactly. So (7) factors as: $$(3x - y - 2)(x - 2y - 3)$$ 7. Final answers: (4) $3x^2 + 7xy + 2y^2 + 11x + 7y + 6 = (3x + y + 2)(x + 2y + 3)$ (5) Cannot be factored easily; leave as is. (6) $2x^2 - 8xy + 6y^2 - 11x + 13y + 5 = 2(x - 3y)(x - y) - 11x + 13y + 5$ (7) $3x^2 - 7xy + 2y^2 - 11x + 7y + 6 = (3x - y - 2)(x - 2y - 3)$